Related papers: Some constructions of Weinstein manifolds with cha…
The dynamics of chaotic billiards is significantly influenced by coexisting regions of regular motion. Here we investigate the prevalence of a different fundamental structure, which is formed by marginally unstable periodic orbits and…
We investigate dynamical properties of chaotic trajectories in mushroom billiards. These billiards present a well-defined simple border between a single regular region and a single chaotic component. We find that the stickiness of chaotic…
We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres.
The collective behavior of a coupled map lattice having {\it unbounded} chaotic local dynamics is investigated through the properties of its mean field. The presence of unstable periodic orbits in the local maps determines the emergence of…
It is an increasingly important problem to study conditions on the structure of a network that guarantee a given behavior for its underlying dynamical system. In this paper we report that a Boolean network may fall within the chaotic…
In this brief note I try to give a simple example of where physical intuition about a collection of interacting qubits can lead to the construction of "natural" versions of what are, generically, quite abstract mathematical objects - in…
This article discusses two recent works by the author, one with Brown and Hurtado on Zimmer's conjecture and one with Bader, Miller and Stover on totally geodesic submanifolds of real and complex hyperbolic manifolds. The main purpose of…
We show how to encode a Weinstein 4-manifold using a multisection diagram with divides, which is a sequence of cut systems on a surface, together with a separating collection of curves. We give two algorithms to construct a multisection…
In this paper, we study geometric rigidity of Riemannian manifolds admitting stable solutions of certain elliptic problems (stability in a variational sense), that is, under suitable hypotheses, we are able to characterize the Riemannian…
Poisson manifolds may be regarded as the infinitesimal form of symplectic groupoids. Twisted Poisson manifolds considered by Severa and Weinstein [math.SG/0107133] are a natural generalization of the former which also arises in string…
In this paper, using pseudo-holomorphic curve method, one proves the Weinstein conjecture in the product $P_1\times P_2$ of two strongly geometrically bounded symplectic manifolds under some conditions with $P_1$. In particular, if $N$ is a…
We study a model inspired by the pinball machine involving chaotic scattering of particles on hard disks with inelasticity. This system exhibits sensitivity not only on the initial conditions of the scattering point particle but also on the…
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…
An outline of the basic Riemannian structures underlying the separation of variables in the Hamilton-Jacobi equation of natural Hamiltonian systems.
The exact elastodynamic scattering theory is constructed to describe the spectral properties of two- and more-cylindrical cavity systems, and compared to an elastodynamic generalization of the semi-classical Gutzwiller unstable periodic…
In this study, the Tauberian Properties of Riesz Means for Sequences of Uncertain Variables were examined, and some interesting properties were revealed.
Recent inelastic neutron scattering experiments on (VO)_2P_2O_7 (VOPO) have seen previously unanticipated sharp peaks in the dynamic structure factor in addition to the pair of triplet modes observed earlier. We argue that the additional…
For partially hyperbolic diffeomorphisms with mostly expanding and mostly contracting centers, we establish a topological structure, called skeleton{a set consisting of finitely many hyperbolic periodic points with maximal cardinality for…
On compact Riemannian manifolds with chaotic geometries, specifically those exhibiting the random wave model conjectured by Berry, we derive heuristically a homogeneous kinetic wave equation that is universal for all such manifolds.
For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality…